On strong Feller property, exponential ergodicity and large deviations principle for stochastic damping Hamiltonian systems with state-dependent switching

This work focuses on a class of stochastic damping Hamiltonian systems with state-dependent switching, where the switching process has countably infinite state space. After establishing existence and uniqueness global weak solution via martingale approach under very mild conditions, paper next proves strong Feller property for regime-switching by killing technique together some resolvent transition probability identities. The commonly used continuity assumption rates qkl(⋅) in literature is relaxed to measurability this paper. Finally provides sufficient conditions exponential ergodicity large deviations principle systems. Several examples van der Pol (overdamped) Langevin are studied detail illustration.